I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, there's a computation of sqrt(2) to 38 decimal places for no discernible purpose. He probably was merely exercising, just like a musician practices scales. Erdős was able to memorise 10 phone numbers at a time from a glance at a phone book, amidst other calculating feats.

I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles. Exercises left to the reader are really exercises in an almost kinesthetic sense.

The computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise. Computation frequently leads to insight. As children are adding two-digit numbers, they start to notice shortcuts about how to perform the computations. These are their own little private theorems, so to speak. So when Devlin here is talking about "mathematical thinking" and the less importance that performing algorithms has today, I hope we don't forget that just because it's less important, it doesn't mean it's not important at all.